First, we note that considering all the possible partitions in \(\Pi \)  (instead of only the \(c+1\)  partitions \(\mathcal {P}_0, \dots , \mathcal {P}_c\) ) will necessarily produce better results than the classical method, and cannot be worst. The number of valid partitions being exponential in general, it is impossible to find efficiently the partition that maximizes an arbitrary quality function. However, we will show in Section  that this is possible with some reasonable assumptions on the quality function \(Q(\mathcal {P})\) . These results are obtained for a general class of additive quality functions that contains the modularity [1]}, the performance [2]} and a new similarity-based quality function introduced in Section .
